Orders at Infinity of Modular Forms with Heegner Divisors

Borcherds described the exponents a(n) in product expansions f = q Q∞ n=1(1−q ) of meromorphic modular forms with a Heegner divisor. His description does not directly give any information about h, the order of vanishing at infinity of f . We give p-adic formulas for h in terms of generalized traces given by sums over the zeroes and poles of f . Specializing to the case of the Hilbert class polynomial f = Hd(j(z)) yields p-adic formulas for class numbers that generalize past results of Bruinier, Kohnen and Ono. We also give new proofs of known results about the irreducible decomposition of the supersingular polynomial Sp(X).